Boundary elements with mesh refinements for the wave equation
For computational acoustics and wave propagation, this provides a rigorous framework for mesh refinement near corners/edges, improving accuracy for engineering problems like tire noise.
The paper develops boundary integral equations for the wave equation in polyhedral domains, showing that graded meshes recover optimal approximation rates for non-smooth solutions. Numerical experiments on screen problems validate the theory, including applications to tire sound emission.
The solution of the wave equation in a polyhedral domain in $\mathbb{R}^3$ admits an asymptotic singular expansion in a neighborhood of the corners and edges. In this article we formulate boundary and screen problems for the wave equation as equivalent boundary integral equations in time domain, study the regularity properties of their solutions and the numerical approximation. Guided by the theory for elliptic equations, graded meshes are shown to recover the optimal approximation rates known for smooth solutions. Numerical experiments illustrate the theory for screen problems. In particular, we discuss the Dirichlet and Neumann problems, as well as the Dirichlet-to-Neumann operator and applications to the sound emission of tires.